Question #42123

A block of mass $m = 1 \text{ kg}$ is attached to a horizontal ideal spring of spring constant $k = 100 \text{ N/m}$. The block oscillates in a fluid where the drag force transitions from linear (Stokes regime) to quadratic (Newtonian turbulent regime) depending on the velocity $v$, modeled as $F_d = -b v - c v |v|$, where $b = \frac{0.4}{\pi} \text{ kg/s}$ and $c = 0.03 \text{ kg/m}$. The block is released from rest at a displacement $A = 1 \text{ m}$. Assuming the energy dissipated per cycle is small compared to the total energy (weak damping approximation), what is the approximate total mechanical energy dissipated by the fluid drag during the first quarter cycle (from $x = A$ to $x = 0$)?